2 Answers
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Transposers have three separate uses: picking up loose items from the environment (e.g. in a mob farm), pulling items out of inventories, and acting as gates/valves for tube networks.

Sucking up items (whether from an inventory or the environment) is done one at a time via a redstone pulse and when not connected to a inventory items will be sucked up from within a 3x3 area around the input face

When powered by a constant redstone signal the transposer will prevent any items from passing through.

Filters can function as either an advanced version of the transposer or as an inline filter in a tube network.

In both cases you can use the filter's internal inventory to determine what items (and what stack size) will be pulled out/allowed through and you can also choose to have all accepted items painted with a single colour. You need to be careful when setting up your tube network with a filter, as tubes will connect to the side of the filter and put items into its inventory.

When pulling from an inventory the filter requires a redstone pulse just like a transposer, but when used inline no pulse is required. A constant signal will also prevent any items from passing through.

The retriever is best thought of as a 'remote filter' as its functionality is the same but with the benefit of not having to be attached to the source of the items it retrieves. This ability also has a downside, because of items in tube networks always going to the closest available inventory it is not guaranteed that an item requested by a retriever will end up passing through the retriever.

Additionally retrievers are the only one of the three blocks to require blulectric power to function.

Transposers only take out a single item at the time so it's only used for very low through put machines, when used inline they can strip the color of an item and when applied with a redstone signal they can block the line. They can also pick up items that land on them and spit it out in the world.

Retrievers are handy on a ore processing line to collect items from the smelters and pulverisers/macerators, they are handy for multiple machines with low throughput. They need power though but a lava + 4 water Thermopile can provide just fine, also note that items pulled out with a retriever will go to the closest retriever that can take the item.

Filters can take out stacks (with nothing in the slots) at the time or filter out what you want to take out or let through when used inline. When they are configured they can also be used to take out a certain amount of an item (like 9 at the time to put into a packager).